Duality Theory in Logic

Transcription

1 Duality Theory in Logic Sumit Sourabh ILLC, Universiteit van Amsterdam Cool Logic 14th December

2 Duality in General Duality underlines the world Most human things go in pairs (Alcmaeon, 450 BC) Existence of an entity in seemingly di erent forms, which are strongly related. Dualism forms a part of the philosophy of eastern religions. In Physics : Wave-particle duality, electro-magnetic duality, Quantum Physics,...

3 Duality in Mathematics Back and forth mappings between dual classes of mathematical objects. Lattices are self-dual objects Projective Geometry Vector Spaces In logic, dualities have been used for relating syntactic and semantic approaches.

4 Algebras and Spaces Logic fits very well in between.

5 Algebras Equational classes having a domain and operations. Eg. Groups, Lattices, Boolean Algebras, Heyting Algebras..., Homomorphism, Subalgebras, Direct Products, Variety,... Power Set Lattice (BDL) BA = (BDL + ) s.t.a _ a =1

6 (Topological) Spaces Topology is the study of spaces. A toplogy on a set X is a collection of subsets (open sets) of X, closed under arbitrary union and finite intersection. R Open Sets correspond to neighbourhoods of points in space. Metric topology, Product topology, Discrete topology... Continuous maps, Homeomorphisms, Connectedness, Compactness, Hausdor ness,... I dont consider this algebra, but this doesn t mean that algebraists cant do it. (Birkho )

7 A brief history of Propositional Logic Boole s The Laws of Thought (1854) introduced an algebraic system for propositional reasoning. Boolean algebras are algebraic models for Classical Propositional logic. Propositional logic formulas correspond to terms of a BA. George Boole ( ) `CPL ', = CPL ', = BA ' = >

8 Representation in Finite case Representation Theorems every element of the class of structures X is isomorphic to some element of the proper subclass Y of X Important and Useful (algebraic analogue of completeness) Cayley s theorem, Riesz s theorem,...

9 Representation in Finite case Representation Theorems every element of the class of structures X is isomorphic to some element of the proper subclass Y of X Important and Useful (algebraic analogue of completeness) Cayley s theorem, Riesz s theorem,... Representation for Finite case (Lindembaum-Tarski 1935) Easy!

10 Representation in Finite case Representation Theorems every element of the class of structures X is isomorphic to some element of the proper subclass Y of X Important and Useful (algebraic analogue of completeness) Cayley s theorem, Riesz s theorem,... Representation for Finite case (Lindembaum-Tarski 1935) Easy! Atoms q FBA i PowerSet Set Map every element of the algebra to the set of atoms below it f (b) ={a 2 At(B) a apple b} for all b 2 B FBA = P(Atoms)

11 Stone Duality A cardinal principle of modern mathematical research maybe be stated as a maxim: One must always topologize. '$ BA &% i '$ q Stone &% -Marshall H. Stone BA = Stone Op Stone s Representation Theorem (1936) M. H. Stone ( )

12 From Spaces to Algebras and back Stone spaces Compact, Hausdor, Totally disconnected eg. 2 A,Cantorset,Q \ [0, 1] Stone space to BA Lattice of clopen sets of a Stone space form a Boolean algebra

13 From Spaces to Algebras and back Stone spaces Compact, Hausdor, Totally disconnected eg. 2 A,Cantorset,Q \ [0, 1] Stone space to BA Lattice of clopen sets of a Stone space form a Boolean algebra BA to Stone space Key Ideas : (i) Boolean algebra can be seen a Boolean ring (Idempotent) (ii) Introducing a topology on the space of ultrafilters of the Boolean ring What is an ultrafilter?

14 From Spaces to Algebras and back A filter on a BA is a subset F of BA such that 1 2 F, 0 /2 F ; if u 2 F and v 2 F,thenu^v 2 F ; if u, v 2 B, u 2 F and u apple v, thenv 2 F. In short, its an upset, closed under meets. An ultrafilter U, is a filter such that either a 2 U or a 2 U. Example: LetP(X ) be a powerset algebra Then the subset " {x} = {A 2 P(X ) x 2 A} is an ultrafilter. Non-principal ultrafilters exist (Axiom of Choice)

15 From Spaces to Algebras and back A filter on a BA is a subset F of BA such that 1 2 F, 0 /2 F ; if u 2 F and v 2 F,thenu^v 2 F ; if u, v 2 B, u 2 F and u apple v, thenv 2 F. In short, its an upset, closed under meets. An ultrafilter U, is a filter such that either a 2 U or a 2 U. Example: LetP(X ) be a powerset algebra Then the subset " {x} = {A 2 P(X ) x 2 A} is an ultrafilter. Non-principal ultrafilters exist (Axiom of Choice) (i) Map an element of B to the set of ultrafilters containing it f (b) ={u 2 S(B) a 2 u} (ii) Topology on S(B), is generated by the following basis {u 2 S(B) b 2 u} where b 2 B

16 From Spaces to Algebras and back A filter on a BA is a subset F of BA such that 1 2 F, 0 /2 F ; if u 2 F and v 2 F,thenu^v 2 F ; if u, v 2 B, u 2 F and u apple v, thenv 2 F. In short, its an upset, closed under meets. An ultrafilter U, is a filter such that either a 2 U or a 2 U. Example: LetP(X ) be a powerset algebra Then the subset " {x} = {A 2 P(X ) x 2 A} is an ultrafilter. Non-principal ultrafilters exist (Axiom of Choice) (i) Map an element of B to the set of ultrafilters containing it f (b) ={u 2 S(B) a 2 u} (ii) Topology on S(B), is generated by the following basis {u 2 S(B) b 2 u} where b 2 B Morphisms and Opposite (contravariant) Duality

17 The Complete Duality Atoms q CBA Set i 6 PowerSet 6 (.) U UlF q BA Stone i Clop Canonical extensions Stone-Céch Compactification

18 Modal logic as we know it Kripke had been introduced to Beth by Haskell B. Curry, who wrote the following to Beth in 1957 I have recently been in communication with a young man in Omaha Nebraska, named Saul Kripke... This young man is a mere boy of 16 years; yet he has read and mastered my Notre Dame Lectures and writes me letters which would do credit to many a professional logician. I have suggested to him that he write you for preprints of your papers which I have already mentioned. These of course will be very di cult for him, but he appears to be a person of extraordinary brilliance, and I have no doubt something will come of it. Saul Kripke

19 Modal logic as we know it Kripke had been introduced to Beth by Haskell B. Curry, who wrote the following to Beth in 1957 I have recently been in communication with a young man in Omaha Nebraska, named Saul Kripke... This young man is a mere boy of 16 years; yet he has read and mastered my Notre Dame Lectures and writes me letters which would do credit to many a professional logician. I have suggested to him that he write you for preprints of your papers which I have already mentioned. These of course will be very di cult for him, but he appears to be a person of extraordinary brilliance, and I have no doubt something will come of it. Saul Kripke Saul Kripke, A Completeness Theorem in Modal Logic. J. Symb. Log. 24(1): 1-14 (1959)

20 Modal logic before the Kripke Era C.I. Lewis, Survey of Symbolic Logic, 1918 (Axiomatic system S1-S5) Syntactic era ( ) Algebraic semantics, JT Duality,... The Classical era ( ) Revolutionary Kripke semantics, Frame completeness,... Modern era (1972- present) Incompleteness results (FT 72, JvB 73), Universal algebras in ML, CS applications,... Modal Algebra (MA) = Boolean Algebra + Unary operator 1. (a _ b) = a _ b 2.? =? 3. (a! b)! ( a! b) (Monotonicity of )

21 Jóhsson-Tarski Duality UltFr q CBAO KR i 6 ComplexAlg 6 (.) U q UltF BAO MS i Clop Jóhnsson-Tarski Duality ( ) Bjarni Jóhnsson Alfred Tarski ( )

22 Modal Spaces and Kripke Frames Key Idea: We already know, ultrafilter frame of the BA forms a Stone space. For BAO, we add the following relation between ultrafilters Ruv i fa 2 u for all a 2 v Descriptive General Frames Unify relational and algebraic semantics DGF = KFr + admissible or clopen valuations Validity on DGF ) Validity on KFr Converse (Persistence) only true for Sahlqvist formulas

23 Algebraic Soundness and Completeness Theorem: Let set of modal formulas. Define V = {A 2 BAO 8'(' 2 ) A = ' = >)} Then for every, `K i V = = >.

24 Algebraic Soundness and Completeness Theorem: Let set of modal formulas. Define V = {A 2 BAO 8'(' 2 ) A = ' = >)} Then for every, `K i V = = >. Soundness: By induction on the depth of proof of.

25 Algebraic Soundness and Completeness Theorem: Let set of modal formulas. Define V = {A 2 BAO 8'(' 2 ) A = ' = >)} Then for every, `K i V = = >. Soundness: By induction on the depth of proof of. Completeness: Assume0 K.Toshow:9A 2 V and A = ( 6= >).The Lindenbaum-Tarski algebra is the canonical witness (Define 0 i `K ( $ 0 ).

26 Algebraic Soundness and Completeness Theorem: Let set of modal formulas. Define V = {A 2 BAO 8'(' 2 ) A = ' = >)} Then for every, `K i V = = >. Soundness: By induction on the depth of proof of. Completeness: Assume0 K.Toshow:9A 2 V and A = ( 6= >).The Lindenbaum-Tarski algebra is the canonical witness (Define 0 i `K ( $ 0 ). Canonical Models are Ultrafilter frames of Lindenbaum-Tarski algebra

27 Algebraic Soundness and Completeness Theorem: Let set of modal formulas. Define V = {A 2 BAO 8'(' 2 ) A = ' = >)} Then for every, `K i V = = >. Soundness: By induction on the depth of proof of. Completeness: Assume0 K.Toshow:9A 2 V and A = ( 6= >).The Lindenbaum-Tarski algebra is the canonical witness (Define 0 i `K ( $ 0 ). Canonical Models are Ultrafilter frames of Lindenbaum-Tarski algebra Disjoint Union $ Product Bounded Morphic image $ Subalgebras Generated subframe $ Homomorphic image

28 PrFr CHA Esakia Duality q IKr i 6 ComplexAlg 6 (.) U PrF q HA Esakia i Op Esakia Duality (1974) Leo Esakia ( ) Useful in characterizing Intermidiate logics.

29 List of Dual Structures in Logic Duality Algebra Space Logic Priestly Duality DL Priestly negation free CL Esakia Duality HA Esakia IPL Stone Dualtiy BA Stone CPL Jóhnsson-Tarski Duality MA MS ML...

30 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,...

31 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,...

32 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable?

33 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm)

34 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary?

35 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary? Van Benthem s Theorem [JvB 76]

36 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary? Van Benthem s Theorem [JvB 76] Which modal formulas define elementary class of frames?

37 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary? Van Benthem s Theorem [JvB 76] Which modal formulas define elementary class of frames? T, 4, B etc define elementary class of frames

38 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary? Van Benthem s Theorem [JvB 76] Which modal formulas define elementary class of frames? T, 4, B etc define elementary class of frames Innocent looking McKinsey ( p! p) does not!

39 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary? Van Benthem s Theorem [JvB 76] Which modal formulas define elementary class of frames? T, 4, B etc define elementary class of frames Innocent looking McKinsey ( p! p) does not! It s an undecidable problem [Chagrova 89]

40 Frame Definability and Correspondence Elementary class of frames Reflexive, Transitive, Antisymmetric,... Modally def. class of frames p! p, p! p, ( p! p)! p,... Which Elementary class of frames are Modally definable? Goldblatt-Thomason Theorem [GT 74] (Duality + Birkho s Thm) Which Modally definable class of frames are elementary? Van Benthem s Theorem [JvB 76] Which modal formulas define elementary class of frames? T, 4, B etc define elementary class of frames Innocent looking McKinsey ( p! p) does not! It s an undecidable problem [Chagrova 89] Sahlqvist formulas provide su cient conditions

41 Correspondence Theory Johan s PhD thesis Modal Correspondence Theory in Correspondence Provides su cient syntactic conditions for first order frame correspondence eg. Sahlqvist formulas. Completeness Sahlqvist formulas are canonical and hence axiomatization by Sahlqvist axioms is complete.

42 Classical Correspondence '$ * Rel.St. I &% Modal logic i Correspondence q First order logic

43 Correspondence via Duality '$ ~ '$ q = Algebras &% i Spaces &% I Algebraic Logic Modal logic i Correspondence Model theory q First order logic

44 Correspondence via Duality Key Ideas (i) Use the properties of the algebra to drive the correspondence mechanism. (ii) Use the (order theoretic) properties of the operators to define sahlqvist formulas Eg. p! p iif p apple p i p apple ( ) 1 p ( as SRA) i 8i 8j i apple p &( ) 1 p apple m i 8i 8j ( ) 1 i apple j i 8x, x 2 R[x] SLR SRA/SLR SRA Sahlqvist formula Advantages (i) Counter-intuitive frame conditons can be easily obtained (eg. Löb s axiom) (ii) The approach generalizes to a wide variety of logics.

45 Point Free Topology Point free Topology Open sets are first class citizens Lattice theoretic (algebraic) approach to topology Sober spaces and Spatial locales Gelfand dualtiy locally KHaus and the C*-algebra of continuous complex-valued functions on X Understanding spaces by maps. Algebraic Topology? Peter T. Johnstonne Israel Gelfand ( )

46 Coalgebras Modal logics are Colagebraic [CKPSV 08] Kripke frames as transition systems. Coalgebraic Stone Duality Ult '$ ~ '$ q = MHom MA &% i Clop Stone &% Vietoris MA = Alg(Clop V Ult) = CoAlg(V) op

47 References Lattices and Order, B.DaveyandH.Priestly Modal Logic, [BRV]Ch.5 Stone Spaces, P. T. Johnstonne Mathematical Structures in Logic

48 The Beatles Within You Without You is a song written by George Harrison, released on The Beatles 1967 album, Sgt. Pepper s Lonely Hearts Club Band.

49 The Beatles... And the time will come when you see We re all one, and life flows on within you and without you

50 The Beatles... And the time will come when you see We re all one, and logic flows on within you and without you (summarizes duality theory quite well!)